Linear equations with two variables in Piatetski-Shapiro sequences

TL;DR

This paper proves that linear equations with two variables are solvable within Piatetski-Shapiro sequences for certain exponents, and analyzes the size of the set of exponents where solvability occurs, including its Hausdorff dimension.

Contribution

It establishes solvability of linear equations in Piatetski-Shapiro sequences for exponents between 1 and 2, and characterizes the Hausdorff dimension of the set of exponents for which solvability holds.

Findings

Linear equations are solvable in PS(α) for 1<α<2.

The set of α where solvability occurs in (s,t) has Hausdorff dimension 2/s.

The Hausdorff dimension of the set is exactly 2/s.

Abstract

For every non-integral $\alpha>1$, the sequence of the integer parts of $n^{\alpha}$ $(n=1,2,\ldots)$ is called the Piatetski-Shapiro sequence with exponent $\alpha$, and let $\mathrm{PS}(\alpha)$ denote the set of all those terms. For all $X\subseteq \mathbb{N}$, we say that an equation $y=ax+b$ is solvable in $X$ if the equation has infinitely many solutions of distinct pairs $(x,y)\in X^2$. Let $a,b\in \mathbb{R}$ with $a\neq 1$ and $0\leq b<a$, and suppose that the equation $y=ax+b$ is solvable in $\mathbb{N}$. We show that for all $1<\alpha<2$ the equation $y=ax+b$ is solvable in $\mathrm{PS}(\alpha)$. Further, we investigate the set of $\alpha \in (s,t)$ so that the equation $y=ax+b$ is solvable in $\mathrm{PS}(\alpha)$ where $2< s <t$. Finally, we show that the Hausdorff dimension of the set is coincident with $2/s$.